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By John Baylis, Rod Haggarty

'...quite the simplest one i've got had the fortune to read...admirable substitute interpreting for a starting place path introducing collage mathematics.' David Tall, the days larger academic complement

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28]. 0 We conclude this chapter with some exercises on equivalence relations. 29] Verify that the relation ~ defined on ~ x ~ by (a, b) ~ (e, d) if and only if a + d = b + e is an equivalence relation. 30] Do the same for (a, b)fYi(e,d) if and only if ad = be, where fYi is a relation on 7L. x (7L. \ {0 }). 3 should recognise the equivalence classes involved in the two examples above! 31] How many different equivalence relations are there on a threeelement set? 32] A relation fYi on {a, b, c, d} consists of the ordered pairs (a, b), (a, c), (a, a), (b, d), (e, c).

3) If O€)Z,X®Z €)y®z if and only if X€)y . (4) If Z€)O,X®Z €)y®z if and only if y €)x. 1 The Construction of Q from 7L We must first say something about notation. 8. This in turn can be written [n, 1], using n for the natural number a - b + 1. 6]. 7], is - [m + 1,1]. Hence, the integers can be partitioned into three distinct classes: { [n + 1, 1]: nE N } - called the positive integers; {[I , I]} -the zero integer; and {[1, n + 1]: nE N } - called the negative integers. To minimise pedantry in notation, we shall revert to the usual ab and a + b and a < b for multiplication, addition and 'less than ' on 71, and agree to write n,O and - n for the integers [n + 1,1], en, n] and [1, n + 1], respectively.

5 a still greater appearance of respectability by attempting to define it (and irrational numbers in general) in terms of rational numbers, just as the rationals were defined in terms of integers and integers in terms of naturals in Chapter 3. By doing this, we aim to make the notion of mathematical existence more precise, but first it is worth thinking about the sense in which irrational numbers definitely do not exist. Let us provisionally accept the evidence (or faith) of modern physics that the universe is constructed entirely from a small number of types of elementary particles with fixed masses, ml> m 2 , • • • , mn • For simplicity assume for the moment that there is just one type, with mass m.

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